A fibered codimension $2$ link in a manifold $M$ can be identified with an open book decomposition of $M$ (the object that is described at the beginning of Bruno Martelli's answer).
The existence of open books was studied in Quinn's paper:
Open book decompositions, and the bordism of automorphisms.
In particular, any odd-dimensional (closed oriented) manifold admits the structure of an open book. In even dimensions there are obstructions. For example, the only surface admitting an open book is $S^2$.
A general obstruction is coming from the signature as Bruno Martelli was pointing out. Given an open book decomposition of $M$, it is relatively easy to construct an orientation reversing diffeomorphisms on the neighborhood of the codimension $2$ manifold and on its complement. Thus the signature has to vanish.
In the above paper, Quinn identifies another (more complicated) algebraic obstruction to the existence of open books in even dimensions larger than $4$ and shows on the other hand that these are the only obstructions.
In dimension $4$ the question about the existence of open book structures seems to be unsolved (apart from that the signature has to vanish). (Quinn's paper uses the $s$-cobordism theorem and thus the argument is probably not working in dimension $4$.)